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Mathematical physics
Mathematical physics is the scientific discipline concerned with the interface of mathematics and physics. There is no real consensus about what does or does not constitute mathematical physics. A very typical definition is the one given by the Journal of Mathematical Physics: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories."Definition from the Journal of Mathematical Physics.http://jmp.aip.org/jmp/staff.jsp This definition does, however, not cover the situation where results from physics are used to help prove facts in abstract mathematics which themselves have nothing particular to do with physics. This phenomenon has become increasingly important, with developments from string theory research breaking new ground in mathematics. Eric Zaslow coined the phrase physmatics to describe these developments,Zaslow E.,Physmatics although other people would consider them as part of mathematical physics proper. Important fields of research in mathematical physics include: functional analysis/quantum physics, geometry/general relativity and combinatorics/probability theory/statistical physics. More recently, string theory has managed to make contact with many major branches of mathematics including algebraic geometry, topology, and complex geometry. Scope of the subject There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. The theory of partial differential equations (and the related areas of variational calculus, Fourier analysis, potential theory, and vector analysis) are perhaps most closely associated with mathematical physics. These were developed intensively from the second half of the eighteenth century (by, for example, D'Alembert, Euler, and Lagrange) until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, elasticity theory, acoustics, thermodynamics, electricity, magnetism, and aerodynamics. The theory of atomic spectra (and, later, quantum mechanics) developed almost concurrently with the mathematical fields of linear algebra, the spectral theory of operators, and more broadly, functional analysis. These constitute the mathematical basis of another branch of mathematical physics. The special and general theories of relativity require a rather different type of mathematics. This was group theory: and it played an important role in both quantum field theory and differential geometry. This was, however, gradually supplemented by topology in the mathematical description of cosmological as well as quantum field theory phenomena. Statistical mechanics forms a separate field, which is closely related with the more mathematical ergodic theory and some parts of probability theory. The usage of the term 'Mathematical physics' is sometimes idiosyncratic. Certain parts of mathematics that initially arose from the development of physics are not considered parts of mathematical physics, while other closely related fields are. For example, ordinary differential equations and symplectic geometry are generally viewed as purely mathematical disciplines, whereas dynamical systems and Hamiltonian mechanics belong to mathematical physics. Prominent mathematical physicists One of the earliest mathematical physicists was the eleventh century Iraqi physicist and mathematician, Ibn al-Haytham 965-1039, known in the West as Alhazen. His conceptions of mathematical models and of the role they play in his theory of sense perception, as seen in his Book of Optics (1021), laid the foundations for mathematical physics. Other notable mathematical physicists at the time included Abū Rayhān al-Bīrūnī 973-1048 and Al-Khazini 1115-1130, who introduced algebraic and fine calculation techniques into the fields of statics and dynamics.Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", p. 642, in The great seventeenth century English physicist and mathematician, Isaac Newton 1642-1727, developed a wealth of new mathematics (for example, calculus and several numerical methods (most notably Newton's method) to solve problems in physics. Other important mathematical physicists of the seventeenth century included the Dutchman Christiaan Huygens 1629-1695 (famous for suggesting the wave theory of light), and the German Johannes Kepler 1571-1630 (Tycho Brahe's assistant, and discoverer of the equations for planetary motion/orbit). In the eighteenth century, two of the great innovators of mathematical physics were Swiss: Daniel Bernoulli 1700-1782 (for contributions to fluid dynamics, and vibrating strings), and, more especially, Leonhard Euler 1707-1783, (for his work in variational calculus, dynamics, fluid dynamics, and many other things). Another notable contributor was the Italian-born Frenchman, Joseph-Louis Lagrange 1736-1813 (for his work in mechanics and variational methods). In the late eighteenth and early nineteenth centuries, important French figures were Pierre-Simon Laplace 1749-1827 (in mathematical astronomy, potential theory, and mechanics) and Siméon Denis Poisson 1781-1840 (who also worked in mechanics and potential theory). In Germany, both Carl Friedrich Gauss 1777-1855 (in magnetism) and Carl Gustav Jacobi 1804-1851 (in the areas of dynamics and canonical transformations) made key contributions to the theoretical foundations of electricity, magnetism, mechanics, and fluid dynamics. Gauss (along with Euler) is considered by many to be one of the three greatest mathematicians of all time. His contributions to non-Euclidean geometry laid the groundwork for the subsequent development of Riemannian geometry by Bernhard Riemann 1826-1866. As we shall see later, this work is at the heart of general relativity. The nineteenth century also saw the Scot, James Clerk Maxwell 1831-1879, win renown for his four equations of electromagnetism, and his countryman, Lord Kelvin 1824-1907 make substantial discoveries in thermodynamics. Among the English physics community, Lord Rayleigh 1842-1919 worked on sound; and George Gabriel Stokes 1819-1903 was a leader in optics and fluid dynamics; while the Irishman William Rowan Hamilton 1805-1865 was noted for his work in dynamics. The German Hermann von Helmholtz 1821-1894 is best remembered for his work in the areas of electromagnetism, waves, fluids, and sound. In the U.S.A., the pioneering work of Josiah Willard Gibbs 1839-1903 became the basis for statistical mechanics. Together, these men laid the foundations of electromagnetic theory, fluid dynamics and statistical mechanics. The late nineteenth and the early twentieth centuries saw the birth of special relativity. This had been anticipated in the works of the Dutchman, Hendrik Lorentz 1853-1928, with important insights from Jules-Henri Poincaré 1854-1912, but which were brought to full clarity by Albert Einstein 1879-1955. Einstein then developed the invariant approach further to arrive at the remarkable geometrical approach to gravitational physics embodied in general relativity. This was based on the non-Euclidean geometry created by Gauss and Riemann in the previous century. Einstein's special relativity replaced the Galilean transformations of space and time with Lorentz transformations in four dimensional Minkowski space-time. His general theory of relativity replaced the flat Euclidean geometry with that of a Riemannian manifold, whose curvature is determined by the distribution of gravitational matter. This replaced Newton's scalar gravitational force by the Riemann curvature tensor. The other great revolutionary development of the twentieth century has been quantum theory, which emerged from the seminal contributions of Max Planck 1856-1947 (on black body radiation) and Einstein's work on the photoelectric effect. This was, at first, followed by a heuristic framework devised by Arnold Sommerfeld 1868-1951 and Niels Bohr 1885-1962, but this was soon replaced by the quantum mechanics developed by Max Born 1882-1970, Werner Heisenberg 1901-1976, Paul Dirac 1902-1984, Erwin Schrödinger 1887-1961, and Wolfgang Pauli 1900-1958. This revolutionary theoretical framework is based on a probabilistic interpretation of states, and evolution and measurements in terms of self-adjoint operators on an infinite dimensional vector space (Hilbert space, introduced by David Hilbert 1862-1943). Paul Dirac, for example, used algebraic constructions to produce a relativistic model for the electron, predicting its magnetic moment and the existence of its antiparticle, the positron. Later important contributors to twentieth century mathematical physics include Satyendra Nath Bose 1894-1974, Julian Schwinger 1918-1994, Sin-Itiro Tomonaga 1906-1979, Richard Feynman 1918-1988, Freeman Dyson 1923-, Hideki Yukawa 1907-1981, Roger Penrose 1931-, Stephen Hawking 1942-, and Edward Witten 1951-. Mathematically rigorous physics The term ' 'mathematical' ' physics is also sometimes used in a special sense, to distinguish research aimed at studying and solving problems inspired by physics within a mathematically rigorous framework. Mathematical physics in this sense covers a very broad area of topics with the common feature that they blend pure mathematics and physics. Although related to theoretical physics, 'mathematical' physics in this sense emphasizes the mathematical rigour of the same type as found in mathematics. On the other hand, theoretical physics emphasizes the links to observations and experimental physics which often requires theoretical physicists (and mathematical physicists in the more general sense) to use heuristic, intuitive, and approximate arguments. Such arguments are not considered rigorous by mathematicians. Arguably, rigorous mathematical physics is closer to mathematics, and theoretical physics is closer to physics. Such mathematical physicists primarily expand and elucidate physical theories. Because of the required rigor, these researchers often deal with questions that theoretical physicists have considered to already be solved. However, they can sometimes show (but neither commonly nor easily) that the previous solution was incorrect. The field has concentrated in three main areas: (1) quantum field theory, especially the precise construction of models; (2) statistical mechanics, especially the theory of phase transitions; and (3) nonrelativistic quantum mechanics (Schrödinger operators), including the connections to atomic and molecular physics. The effort to put physical theories on a mathematically rigorous footing has inspired many mathematical developments. For example, the development of quantum mechanics and some aspects of functional analysis parallel each other in many ways. The mathematical study of quantum statistical mechanics has motivated results in operator algebras. The attempt to construct a rigorous quantum field theory has brought about progress in fields such as representation theory. Use of geometry and topology plays an important role in string theory. The above are just a few examples. An examination of the current research literature would undoubtedly give other such instances. Notes References * * Bibliographical references The classics :* :* :* :* (pbk.) :* (softcover) :* :* :*''This is a'' reprint'' of the second (1980) edition of this title.'' :* :*''This is a reprint of the'' 1956 second edition. :* :*''This is a'' reprint'' of the original (1953) edition of this title.'' :* :* :* :*''This tome was reprinted in 1985. :* :* :* Textbooks for undergraduate studies :* (pbk.) :* :* :* :* :* (set : pbk.) Other specialised subareas :* :* (pbk.) :* (pbk.) :* :* :* (pbk.) See also * Important publications in Mathematical Physics * Theoretical physics External links * Communications in Mathematical Physics * Journal of Mathematical Physics * Mathematical Physics Electronic Journal * International Association of Mathematical Physics * Erwin Schrödinger International Institute for Mathematical Physics * [http://eqworld.ipmnet.ru/en/solutions/lpde.htm Linear Mathematical Physics Equations: Exact Solutions] - from EqWorld * [http://eqworld.ipmnet.ru/en/solutions/eqindex/eqindex-mphe.htm Mathematical Physics Equations: Index] - from EqWorld * [http://eqworld.ipmnet.ru/en/solutions/npde.htm Nonlinear Mathematical Physics Equations: Exact Solutions] - from EqWorld * [http://eqworld.ipmnet.ru/en/methods/meth-pde.htm Nonlinear Mathematical Physics Equations: Methods] - from EqWorld Category:Mathematical science occupations